# 2008 AMC 12B Problems/Problem 1

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## Problem

A basketball player made $5$ baskets during a game. Each basket was worth either $2$ or $3$ points. How many different numbers could represent the total points scored by the player?

$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

## Solution 1

If the basketball player makes $x$ three-point shots and $5-x$ two-point shots, he scores $3x+2(5-x)=10+x$ points. Clearly every value of $x$ yields a different number of total points. Since he can make any number of three-point shots between $0$ and $5$ inclusive, the number of different point totals is $6 \Rightarrow E$.

### Solution 2

Stars and bars can also be utilized to solve this problem. Since we need to decide what number of 2's and 3's are scored, and there are a total of 5 shots. It can be written like such: _ _ _ | _ _. Solving this, we get $6 \Rightarrow E$.